Analyse en Terminale : Dérivées, Intégrales et Limites

Question 1 sur 10 10:00

[{"id":79792,"question":"Quelle est la dérivée de la fonction f(x) = 3x² + 2x - 5 ?","option_a":"6x + 2","option_b":"3x² + 2","option_c":"6x - 5","option_d":"x² + 2x","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de 3x² est 6x, celle de 2x est 2, et celle de -5 est 0. Donc f'(x) = 6x + 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79793,"question":"La limite de f(x) = (x² - 1)\/(x - 1) lorsque x tend vers 1 est égale à 2.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"En simplifiant, f(x) = (x - 1)(x + 1)\/(x - 1) = x + 1 pour x ≠ 1. Donc lim(x→1) f(x) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79794,"question":"Quelle est la primitive de f(x) = 4x³ ?","option_a":"x⁴ + C","option_b":"12x² + C","option_c":"x⁴\/4 + C","option_d":"4x⁴ + C","option_e":"","option_f":"","bonne_reponse":"C","explication":"La primitive de xⁿ est xⁿ⁺¹\/(n+1). Donc ∫4x³ dx = 4x⁴\/4 + C = x⁴ + C.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79795,"question":"L'intégrale de f(x) = 2 entre 0 et 3 est égale à 6.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"L'intégrale de 2 entre 0 et 3 est 2*(3-0) = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79796,"question":"Quelle est la dérivée de la fonction f(x) = e^(2x) ?","option_a":"2e^(2x)","option_b":"e^(2x)","option_c":"2x e^(2x)","option_d":"e^(x)","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de e^(u(x)) est u'(x)e^(u(x)). Ici u(x) = 2x, donc u'(x) = 2. D'où f'(x) = 2e^(2x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79797,"question":"La fonction f(x) = x³ - 3x² + 2 admet-elle un extremum local en x = 1 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"f'(x) = 3x² - 6x. f'(1) = 3 - 6 = -3 ≠ 0. Donc pas d'extremum en x = 1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79798,"question":"Quelle est la valeur de l'intégrale ∫(3x² + 2x) dx entre 0 et 1 ?","option_a":"2","option_b":"1","option_c":"4\/3","option_d":"3","option_e":"","option_f":"","bonne_reponse":"A","explication":"∫(3x² + 2x) dx = x³ + x² + C. Entre 0 et 1 : (1 + 1) - (0 + 0) = 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79799,"question":"La fonction f(x) = ln(x) est définie pour x \u003E 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La fonction logarithme népérien ln(x) est définie uniquement pour x \u003E 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79800,"question":"Quelle est la limite de f(x) = (1 - cos(x))\/x lorsque x tend vers 0 ?","option_a":"0","option_b":"1","option_c":"2","option_d":"∞","option_e":"","option_f":"","bonne_reponse":"B","explication":"En utilisant le développement limité de cos(x) = 1 - x²\/2 + o(x²), on obtient (1 - (1 - x²\/2))\/x = x\/2 → 0. Mais en réalité, la limite est 0 (car (1 - cos(x))\/x ≈ x\/2 → 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":79801,"question":"La dérivée de f(x) = sin(3x) est f'(x) = 3cos(3x).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La dérivée de sin(u(x)) est u'(x)cos(u(x)). Ici u(x) = 3x, donc u'(x) = 3. D'où f'(x) = 3cos(3x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]